The Larmor Proposition that Accelerated Point Charges
Radiate Electromagnetic Waves is
Irrelevant in a World of Spatially Distributed Charges

Background

The Larmor Formula

In 1897 the Irish physicist Joseph J. Larmor published an article, "On a dynamical theory of the electric
and luminiferous medium", in the
Philosophical Transactions of the Royal Society, vol. 190, (1897) pp. 205–300. (Third and last in a series of papers with the same name).
In that article Larmor derived a formula for the
power R radiated by a charge of magnitude q accelerating at a rate α

R = ⅔q²α²/c³

where c is the speed of light. This version of the Larmor formula is for the centimeter-gram-second
system of units.

Larmor's analysis
in 1897 was followed by a more comprehensive analysis by Alfred-Marie Liénard in 1898 and,
independently, by Emil Wiechert in 1900. Their analyses were compatible with Einstein's Theory
of Special Relativity published in 1905. For velocities small compared to the speed of light the
formula of Liénard and Wiechert reduce to the Larmor formula.

Larmor's analyses was based upon a prior analysis by Hendrik Lorentz.
But Lorentz' analysis was flawed by a dependence upon there being
luminiferous ether prevading space. Supposedly the effect arises because of the forces resulting from
a charged particle dragging its electric field through the ether. The notion of an ether
was discredited but the proposition continued to be accepted. For convenience of reference
the notion that an accelerated charge emits electromagnetic radiation will sometimes
be referred to as the Larmor Proposition.

A modern derivation of the proposition that accelerated charges radiate electromagnetic
is the one given in the venerable text Classical Electrodynamics by John David Jackson.

Jackson opens his chapter on Radiation by Moving Charges with a very strong statement

It is well known that accelerated charges emit electromagnetic radiation.

Perhaps it would be more proper to say that it is widely believed that accelerated charges under some circumstances emit
electromagnetic radiation. The issue is whether under all circumstances, macroscopic and microscopic, and
in the absence of magnetic fields
accelerated charges emit electromagnetic radiation.

The analysis that Jackson bases on the Liénard-Wiechert potentials depends intrinsically
on the so-called Dirac delta function. This is a "function" that is everywhere zero except at a
point where it is infinite. It is a spike that is construed as the derivative of a step function. Mathematically the delta function is not a function; it is called a
distribution. But Jackson's derivation not only utilizes the delta function but the derivative
of the delta function; a positive spike combined immediately with a negative spike. This is in effect
the second derivative of a step function. It is surprising that such esoteric constructions
as the delta function and its derivative are needed for the proof. Undoubtedly this part of Jackson's
derivation can made into rigorous mathematics.

But Jackson then goes on to say

The instaneous energy flux is given by the Poynting vector

S= (c/4π)(E×B)

This means that the power radiated … is …

Jackson does not explicitly say that the power radiated from an accelerated charge is electromagnetic
waves but that is what he is assuming. However from the analysis of the
proof of the Poynting Theorem it
is known that there is not necessarily any electromagnetic waves involved for the Poynting vector. The electric and magnetic fields
may move and in taking their energy with them generate an energy flow. Thus Jackson fails to prove that
an accelerating charge radiates electromagnetic waves.
So a standard source in electromagnetic analysis has a failed proof. It may be that a rigorous proof
can be constructed but it would be for a point charge.

The Skepticism Among Physicists Concerningthe Proposition that Accelerated Charges Radiate

Sir James Jeans in his book, The Mathematical Theory of
Electricity and Magnetism, published in 1933, says

It must be added that the new dynamics referred to … seems to throw doubt
on this formula for the emission of radiation. Many physicists now question whether any emission of radiation
is produced by the acceleration of an electron, except under certain special conditions.

Richard Feynman in his Lectures on Gravitation says "we have inherited a prejudice that an
accelerating charge should radiate." He argues that the Larmor formula giving the power radiated by an
accelerating charge as proportional to the square of the acceleration "has led us astray." Feynman
maintains that a uniformly accelerating charge does not radiate at all. He argues that it is the
rate of change of acceleration that results in electromagnetic radiation from charged particles.

In the Theory of General Relativity there is what is called The Equivalence Principle. This
is the assertion that there should be no difference between a body at rest experiencing a uniform
gravitational field and a body experiencing uniform acceleration. There is no reason to expect a
charged particle resting in a uniform gravitational field to be emitting radiation. Therefore,
according to the Equivalence Principle there should be no radiation from a charged particle experiencing
uniform acceleration. Something must be
wrong.

The proposition of accelerated charges radiating electromagnetic waves seems to have a nusiance
value; i.e., confronting any atomic or nuclear model involving curved trajectories with the argument
that they should collapse. It is always presumed that the proposition concerning accelerated
charges is correct and therefore the model must be wrong.

But the fact remains that nuclei rotate and yet there is no radiation emitted and they do not collapse.
See for example Fast Nuclear Rotation by Zdzislaw Szymanski. Aage Bohr and
Ben R. Mottelson in their Collective Model given in their two volume study Nuclear Structure conclude cautiously that the empirical
evidence is consistent with nuclear rotation. So the proposition fails empirically. So something must
be wrong with the proposition.

In contrast to Jackson's wholehearted endorsement of the proposition the following sample of eight
texts on electricity and magnetism have no reference to accelerated charges or the Larmor formula
in their indices:

The Effect of Continuously Distributed Charges as Opposed to Point Charges

Suppose a charge of q is considered as
two ½q charges. This would mean that the energy radiation is 2(q²/4)= q²/2.

If the charge of q is considered to be M charges of q/M each then the energy radiated by one
is proportional to (q/M)² and the total radiation is proportional to M(q/M)²=q²/M,
(1/M) of energy radiation by a charge of q. As M increases without bound the energy radiated
goes to zero.

Altogether the effects of the M portions is
1/M of the radiation of a charge of q. Thus if space is infinitely divisible and the charge is distributed
over space like a ball or a sphere there would be an "infinite" number of
infinitesimal pieces and their total radiation would be zero.

E = limitM→∞ (2/3)α²q²/M = 0

Hence the radiation of electromagnetic waves by accelerated charges
would be valid only for point particles and real particles are not point
particles. They have charge distributions over space. They cannot be point particles because
charged point particles would have infinite energy. None could have been created in the Universe.

Thus if space is infinitely divisble then no matter how large is the acceleration the radiation generated by a spatially distributed charge
is zero. Likewise no matter
how large is the charge or how small is the scale so long as the particle is not a point particle
the electromagnetic wave generation is zero.

This is the essential result. What follows is a number of investigations of possible
modifications of this result.

What If Space is not Infinitely Divisible

The ancient Greeks speculated that there is a limit to how small matter can be divided which they
call an a (not) tom (cut), atom. This was a couple of millennia years later proven to be true. Electric
charge was also found to be quantized. It is therefore very plausible that there is an atom or quantum
of length and also of time. Let ε be the quantum of length and δ the quantum of time.
(It is widely presumed that ε=cδ, where c is the speed of light.)

The quantum of spatial volume would then be ε³. If V is the volume of a quantum
of charge then the number of pieces M that charge would be divided into is given by

M = V/ε³

The charge of a proton is distributed over a sphere of radius 0.84 fermi (8.4×10-16
meter). The volume occupied by the charge is then 2.483×10-45 m³.
Some believe that the quantum of length is the Planck length of 1.6162×10-35 meters.
This would make the quantum of volume equal to 4.222×10-105.
If that is the case then

M = 2.483×10-45/ 4.222×10-105 = 4.222×10-105
= 0.588×1060

This would mean that a quantum of charge is divided up into
about 1060 pieces; i.e., M is 1060. This makes the radiation for an
accelerated charge very small unless its acceleration is extremely large.

For the Bohr model of a hydrogen atom the tangential velocity of an electron in the lowest
orbit is 2.19×106 m/s, which is less than 1 percent of the speed of light.

The centripetal acceleration of an electron in a hydrogen atom is 9.13×1022 m/s² and this
squared is 8.357×1045 m²/s4 or 8.357×1049 cm²/s4
The charge of an electron is 1.602×10-19 coulombs. The rate of radiant energy generation
due to the centripetal acceleration of the electron according to the Larmor formula (in cgs units) modified
to take into account the spatial distribution of charge into 0.588×1060
pieces is

This however
is the rate of generation of energy. The amount generated in a Planck time of 5.4×10-44 seconds
is 4.86×10-134 joules, practically speaking nothing. An atom is a quantized system so its
energy cannot change by just any amount. It has to change by the emission
of a photon equal in energy to a quantum of energy of the system. Since there is no provision for
accumulating energy no photon is ever emitted.

The Results for Other Values of the Quanta of Length and Time

The largest the quantum of length could be is the charge diameter of a proton, which is
1.68×10-15 meters.
This would make M equal to 1 and the quantum of time equal to 5.6×10-24 sec.
This would make the rate of radiant energy generation equal to 5.3×10-31 joules/sec.
The energy generated in one quantum of time would then be 3.0×10-54 joules,
again nothing, practically speaking.

The Situation for Acceleration Spikes Due to Collisions of Molecules

The previous analysis above demonstrated the irrelevance of the Larmor proposition for the centripetal
acceleration of electrons in atoms. There remains the possibility that a high level of deceleration or
acceleration due to the collisions of molecules might overcome the other factors in the Larmor
equation to give numerically significant results. Such is not the case as will be demonstrated later.
If the quantum of time is small the magnitude of the acceleration resulting from a reversal of velocity
is large. But if the quantum of time is small then so is the quantum of length and hence the quantum
of volume. The number of pieces M that a charge is divided into is large. The effect of M more than
offsets the increase in the magnitude of acceleration due to a smaller time step. But before the
numerical details are presented it is necessary to compute the energy of photons involved in
thermal radiation.

The Wavelength of Thermal Radiation from H2 at Room Temperature

Wien's Displacement Law says that the most frequent wavelength λmax of radiation for a blackbody at
absolute temperature (°K) of T is

λmax =b/T

where b is equal to 0.0029 mK. Thus for a temperature of 300° K the most frequent wavelength of
thermal radiation is about 10-5 meters. The energy of a photon of this radiation is

Determination of the Radiant Energy Generated
by the Collision of A Charge

As present previously, the Lamor formula in cgs units is

R = (2/3)α²q²/c³

where α is acceleration in cm/s², q is charge and c is the speed of light in cm/s. This is the rate of energy generation.

The acceleration generated in a collision is the change in velocity divided by the time involved in the collision.

The charge diameter of a proton is approximately 1.68 fermi = 1.68×10-13 cm. If this is
taken to be the quantum of length then the quantum of time equal to 5.6×10-24 seconds.

The average velocity of an H2 molecule at room temperature (300deg; K) is about
2×105 cm/s. In a head-on collision with a container wall the direction is reversed.
If this takes place
in one quantum of time the magnitude of the maximum acceleration is

αmax = (4×105/5.6×10-24)
=7.14×1028 cm/sec²

.
This quantity squared is 5.1×1057/sup> cm²/sec4.

The charge of the proton is 1.6×10-19 coulombs. This squared is
2.56×10-38. The value of R is the rate of energy generation; the quantity of energy generated in
one quantum of time δ is then

The energy of a photon generated at room temperature by a head-on collision of an H2
molecule with a container wall of 6.74×10-33 joules
is a far cry from the energy of 2.0×10-21 joules which is the average photon energy
of the thermal radiation from H2 at
room temperature (300° K). The acceleration/deceleration photon energy is about a hundred trillion
times smaller than the thermal radiation photon
energy at room temperature.

Thus if the transformation of thermal kinetic energy into thermal radiation due to collisions is involved in thermal radiation
it
is quantitatively insignificant.
However there are other reasons to believe the accelerated charge effect is even less significant than the
above computation indicates.
The spatial distribution of charge results in the charge being divided up into a myriad of small pieces
having near infinitesimal total effect.

Cyclotron and Synchroton Radiation

A beam of charged particles in a cyclotron or a synchroton is held in a circular orbit by a
powerful magnetic field. Such beams do radiate electromagnetic waves in the visible range.
But radiation from a charge traveling in a magnetic field is a different matter from the Larmor
effect which is supposed take place even in the absence of a magnetic field. Therefore
cyclotron and synchroton radiation do not constitute empirical verification of the Larmor effect.

Conclusion

The original analyses of Larmor, Liénard and Weichert upon which the proposition that an
accelerated point charge radiates electromagnetic
waves were invalid because they were based upon the existence of luminiferous ether. The modern
derivation of J.D. Jackson is also invalid because it relies upon a misinterpretation of the Poynting
vector of the Poynting Theorem.

But even if a valid derivation exists it would be for a
point charge. When a charge is distributed over space the effect
is divided by the number of pieces the charge can be partitioned into.
If space is infinitely divisible the effect is reduced to zero.

If quanta
of length and time exist the effect may be vanishingly small quantitatively.
Therefore the proposition that an accelerated charge radiates
electromagnetic waves may be irrelevant for a physical world of
spatially distributed charges.

Computation were carried out using two values for the quanta of length and
time. One set of computations uses the Planck length and time and the other uses
the charge diameter of a proton as the quantum of length with the quantum of
time being the time required for light to travel that distance.
Those computations of the energy generated from the Larmor effect in one
quantum of time due to the centripetal acceleration of an electron in a
hydrogen atom show the effect to be vanishingly small.

Another set of computations were made for the accelerations and
decelerations due to molecular collisions of H2 gas at room temperature.
The energy of the photons generated by such collisions would be vanishingly
small compared to the average photon energy due to thermal radiation from
that gas.

Thus it is not surprising that there is no evidence for the existence of the
Larmor effect at the atomic level. At the macro level the existence of
cyclotron and synchroton radiation is often taken as evidence of the
Larmor effect. But cyclotron and synchroton radiation involve charged particles
traveling through a magnetic field. That is the only way circular orbits can
be achieved for charged particles. Thus those phenomena are not evidence
for a pure Larmor effect.

Therefore the proposition that an accelerated point charge should radiate
electromagnetic waves is irrelevant for the spatially distributed charges
of the physical world.

If the charge is not a point particle but instead a charge distributed over space then it
might appropriately be considered an infinite number of infinitesimal charges and hence there would be no
electromagnetic waves radiated.